Since you're talking about making delivery/deployment decisions based on the results provided by your script(s) you need to communicate with the team(s) responsible with maintaining the official CI/CD pipeline implementation for your organisation.
If they're using Jenkins and a particular pipeline implementation (multiple technological approaches are possible) compatible with the way you hooked up your script into Jenkins locally they may be able to directly use your approach.
Even if that's not directly possible, there probably still are ways of including your tests in the CI/CD pipeline, only with a more or less different implementation.
Note that there may be related technical and/or logistic questions/requirements that might need to be addressed depending on your organisation particularities, for example: ownership, development/maintenance process and policies, SLAs for the tests, script(s) and/or the resources for their execution.
Even if you don't get your tests integrated at first, you can still run them for every repository commit using your private Jenkins setup and make the results available for everyone to see. In time, if those results prove useful to others they may support your initiative and help with a reconsideration of the initial decision.